The recent discovery of high-temperature superconductivity in iron-based compounds has attracted much attention. How to further increase the superconducting transition temperature (T(c)) and how to understand the superconductivity mechanism are two prominent issues facing the current study of iron-based superconductors. The latest report of high-T(c) superconductivity in a single-layer FeSe is therefore both surprising and significant. Here we present investigations of the electronic structure and superconducting gap of the single-layer FeSe superconductor. Its Fermi surface is distinct from other iron-based superconductors, consisting only of electron-like pockets near the zone corner without indication of any Fermi surface around the zone centre. Nearly isotropic superconducting gap is observed in this strictly two-dimensional system. The temperature dependence of the superconducting gap gives a transition temperature T(c)~ 55 K. These results have established a clear case that such a simple electronic structure is compatible with high-T(c) superconductivity in iron-based superconductors.